Mathematics

The key is to recognize that $f(\theta) \geq 0$ for all $\theta$ means the trigonometric expression has a global minimum of at least zero. By strategically choosing specific values of $\theta$—particularly those that isolate or amplify certain terms—you can extract constraints on the coefficients. The double-angle terms $\cos 2\theta$ and $\sin 2\theta$ can be rewritten using $\cos^2\theta$ and $\sin^2\theta$, creating a connection between first and second-order terms that forces bounds on both $(a,b)$ and $(A,B)$.

The key is to recognize that $f(\theta) \geq 0$ for all $\theta$ means the trigonometric expression has a global minimum of at least zero. By strategically choosing specific values of $\theta$—particularly those that isolate or amplify certain terms—you can extract constraints on the coefficients. The double-angle terms $\cos 2\theta$ and $\sin 2\theta$ can be rewritten using $\cos^2\theta$ and $\sin^2\theta$, creating a connection between first and second-order terms that forces bounds on both $(a,b)$ and $(A,B)$.